How far is CS from being Morse-Bott? If you consider $SU(2)$ connections then for Seifert manifolds the character variety can have quadratic singularities, so it's not a manifold. There are examples of hyperbolic manifolds so that the $SU(2)$ character variety has cubic singularities. In fact, for $SO(3)$ flat connections over 3-manifolds the situation is much worse and you can have any singularity over ${\mathbb Z}$. I suspect the same happens even in $SU(2)$ case but it's harder to prove. So, perturbation to a Morse function is the only way to go.